User:Inthar/MV3: Difference between revisions
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''To be continued...'' | ''To be continued...'' | ||
=== | === Proof of the alternating-gens property === | ||
We now prove that except in the case XYXZXYX, if the scale is pairwise well-formed, the scale is generated by alternating generators. | We now prove that except in the case XYXZXYX, if the scale is pairwise well-formed, the scale is generated by alternating generators. | ||
Assume the scale word S is not multiperiod. To eliminate XYXZXYX we manually check all words up to length 7... (todo) | Assume the scale word S is not multiperiod. To eliminate XYXZXYX we manually check all words up to length 7... (todo) | ||
Now assume len(S) >= 8 | Now assume len(S) >= 8 | ||
=== Proof of "aX bY bZ" === | |||
== MV3 Theorem 2 == | == MV3 Theorem 2 == | ||
''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ||